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Let $k1$ and $k2$ be the principal curvatures of an oriented surface S and N be the field of unit normal vectors to S. Let $F_i : S \mapsto \Bbb R^3$ be the map defined as follows: $F_i(p) = p + \frac 1{k_i}N(p)$.

a) Prove that if $p_0$ is a non-umbilical point and the direction derivative of $k_i, i = 1, 2$ at $p_0$ in the corresponding principal direction does not vanish, then there exists a neighborhood $U$ of $p_0$ of $S$ such that $F_1(U)$ and $F_2(U)$ are regular surfaces.

For this one, I want to use the fact that $k1$ and $k2$ are nowhere zero to prove.

b) Prove that if $\alpha(t)$ is a line of curvature on $U$, which is tangent to the principal direction corresponding to the principal curvature $k1$, then after an appropriate reparametrization the curve $F_1(\alpha(t))$ is a geodesic on the surface $F_1(U)$.

If we assume that I'm able to correctly prove the initial component (that $F_1$ is a regular srface), how can I use the fact that $\alpha(t)$ is a line of curvature on $U$ to prove that, when plugged into the regular surface $F_1(U)$, the curve is geodesic? I'm thinking I can use this information to prove that the field of tangents is parallel along the curve $\alpha(t)$.

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  • $\begingroup$ Please proofread and verify the definition of the mappings $F_i$. What is the coefficient of $N(p)$ supposed to be? Please show us what you've tried doing and ask specific questions. I doubt the linked source gives insight to the best way to approach this problem. $\endgroup$ – Ted Shifrin Dec 7 '18 at 22:15
  • $\begingroup$ @TedShifrin I tried to make my question more specific and fixed a typo with the N(p) coefficient, apologies for that. Let me know if it needs to be more specific. I think I can probably prove part a), it is part b) that I am struggling with. $\endgroup$ – jman63 Dec 7 '18 at 22:38
  • $\begingroup$ Again, you don't indicate what techniques or knowledge you have. I would approach this with the method of moving frames, not classical stuff like in your link. The key thing for b) is to realize what vector $v$ is tangent to the curve $F_1\circ\alpha$, what vectors span the tangent plane of $F_1(U)$, and in what direction the derivative of the tangent vector $v$ is pointing. $\endgroup$ – Ted Shifrin Dec 7 '18 at 22:46
  • $\begingroup$ @TedShifrin - sorry about that, the truth of the matter is I'm not well versed and don't want to say something completely incorrect. I apologize for my ineptitude! However, I appreciate the hints you've given me and I'll look into the moving frame approach further. Thank you for the help! $\endgroup$ – jman63 Dec 7 '18 at 23:05
  • $\begingroup$ Otherwise choose a parametrization of your original surface so that the coordinate curves are lines of curvature. You might also want to check out my differential geometry text (freely available from the link in my profile). $\endgroup$ – Ted Shifrin Dec 7 '18 at 23:26

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